Systems, Methods and Metrics for Wafer High Order Shape Characterization and Wafer Classification Using Wafer Dimensional Geometry Tool

ABSTRACT

Systems and methods for improving results of wafer higher order shape (HOS) characterization and wafer classification are disclosed. The systems and methods in accordance with the present disclosure are based on localized shapes. A wafer map is partitioned into a plurality of measurement sites to improve the completeness of wafer shape representation. Various site based HOS metric values may be calculated for wafer characterization and/or classification purposes, and may also be utilized as control input for a downstream application. In addition, polar grid partitioning schemes are provided. Such polar grid partitioning schemes may be utilized to partition a wafer surface into measurement sites having uniform site areas while providing good wafer edge region coverage.

TECHNICAL FIELD

The disclosure generally relates to the field of wafer surfacemetrology, and particularly to systems and methods for wafer high ordershape characterization and wafer classification.

BACKGROUND

Thin polished plates such as silicon wafers and the like are a veryimportant part of modern technology. A wafer, for instance, may refer toa thin slice of semiconductor material used in the fabrication ofintegrated circuits and other devices. Other examples of thin polishedplates may include magnetic disc substrates, gauge blocks and the like.While the technique described here refers mainly to wafers, it is to beunderstood that the technique also is applicable to other types ofpolished plates as well. The term wafer and the term thin polished platemay be used interchangeably in the present disclosure.

Generally, certain requirements may be established for the flatness andthickness uniformity of the wafers. The semiconductor industry uses thetwo global wafer shape metrics, bow and warp, to describe the overallwafer shape. Global surface fitting using the Zernike polynomials orTaylor polynomials have also been used to describe the wafer shapecomponents.

However, the two global wafer shape metrics, bow and warp, do not havethe required spatial resolution and sensitivity for the local wafershape characterization. Methods based on the whole wafer surface fittingcannot provide the information about the location of wafer local higherorder shape components and often do not have good shape sensitivity evenwith very high polynomial fitting orders.

Therein lies a need for systems, methods and metrics for wafer highorder shape characterization and wafer classification without theaforementioned shortcomings.

SUMMARY

The present disclosure is directed to a method for inspecting a wafer.The method may include: defining a wafer partitioning scheme; obtaininga wafer surface image; partitioning the wafer surface image into aplurality of measurement sites according to the wafer partitioningscheme; calculating a plurality of measurement metrics for each of theplurality of measurement sites based on the acquired wafer surfaceimage; and reporting the plurality of measurement metrics calculated foreach of the plurality of measurement sites in a graphicalrepresentation.

A further embodiment of the present disclosure is directed to a systemfor inspecting a wafer. The system may include an optical systemconfigured for obtaining a wafer surface image. The system may alsoinclude a site based high order wafer shape analysis module incommunication with the optical system. The site based high order wafershape analysis module may be configured for: defining a waferpartitioning scheme; partitioning the wafer surface image into aplurality of measurement sites according to the wafer partitioningscheme; calculating a plurality of measurement metrics for each of theplurality of measurement sites based on the acquired wafer surfaceimage; and reporting the plurality of measurement metrics calculated foreach of the plurality of measurement sites in a graphicalrepresentation.

An additional embodiment of the present disclosure is directed to polargrid partitioning method for partitioning a wafer surface. The methodmay include: specifying a number of sectors and a number of zonesrequired for the polar grid partitioning; calculating a sector angularspan based on the number of sectors specified; calculating a radial spanfor each of the number of zones, wherein the radial span for a firstzone having a first radial distance to the center of the wafer isdifferent from the radial span for a second zone having a second radialdistance to the center of the wafer; and partitioning the wafer surfaceinto a plurality of sites based on the sector angular span and theradial span for each zone, wherein the plurality of sites have uniformsite areas.

An additional embodiment of the present disclosure is directed to polargrid partitioning method for partitioning a wafer surface. The methodmay include: specifying a number of zones K required for the polar gridpartitioning and a number of angular segments M in a center region ofthe wafer; calculating a radial zone length L based on the number ofzones specified; calculating an angular span θ_(i) for the radial zone,wherein i=1, 2, 3, . . . K; and partitioning the wafer surface into aplurality of sites based on the radial zone length L and the angularspan θ for each radial zone, wherein the plurality of sites have uniformsite areas.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory onlyand are not necessarily restrictive of the present disclosure. Theaccompanying drawings, which are incorporated in and constitute a partof the specification, illustrate subject matter of the disclosure.Together, the descriptions and the drawings serve to explain theprinciples of the disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

The numerous advantages of the disclosure may be better understood bythose skilled in the art by reference to the accompanying figures inwhich:

FIG. 1 is a flow diagram illustrating a site based high order shapeanalysis method;

FIG. 2 is an illustration depicting a Cartesian grid partition scheme;

FIG. 3 is an illustration depicting a polar grid partition scheme;

FIG. 4 is an illustration depicting a polar grid partition schemeconfigured for providing uniform measurement site areas;

FIG. 5 is an illustration depicting another polar grid partition schemeconfigured for providing uniform measurement site areas;

FIG. 6 is an illustration depicting site image shape data obtained foreach measurement site;

FIG. 7 is an illustration depicting a site shape image, two surfacefitting images and the corresponding deviation (residue) images;

FIG. 8A is an illustration depicting an exemplary wafer shape image;

FIG. 8B is an illustration depicting the X profile of the exemplarywafer shape image of FIG. 8A;

FIG. 9 are illustrations depicting various graphical representations ofthe site based high order shape analysis in accordance with the presentdisclosure;

FIG. 10A is an illustration depicting a deviation map for site basedhigh order shape analysis;

FIG. 10B is an illustration depicting a second order coefficient derivedmap for site based high order shape analysis;

FIG. 11A is an illustration depicting another exemplary wafer shapeimage;

FIG. 11B is an illustration depicting the profile of the exemplary wafershape image of FIG. 11A in −45° orientation;

FIGS. 12-13 are illustrations depicting various graphicalrepresentations of the site based high order shape analysis inaccordance with the present disclosure;

FIG. 14 is an illustration depicting various graphical representationsof the site based high order shape analysis, presented in polar space,in accordance with the present disclosure;

FIG. 15A is an illustration depicting high order shape metrics obtainedusing a pixel-based shape-slope computation process;

FIG. 15B is an illustration depicting high order shape metrics obtainedusing a polynomial fitting process;

FIG. 16 is a flow diagram illustrating an intra-field data analysismethod;

FIG. 17 is an illustration depicting a number of pre-determined targetlocations within a lithography field and across multiple fields within awafer;

FIG. 18 is an illustration depicts a wafer lithography process relativeto a timeline;

FIG. 19 is an illustration depicting the relationship between the amountof nitride thinning experienced as the magnitude of wafer lip increases;

FIG. 20 is a flow diagram illustrating utilizing the site based highorder shape analysis to control a Chemical Mechanical Planarization orPolishing (CMP) process;

FIG. 21 is an illustration depicting contact gap(s) before and afterchucking;

FIG. 22 is an illustration depicting the observed correlation betweencontact gap and site based shape curvature metrics;

FIG. 23 is a flow diagram illustrating utilizing the site based highorder shape analysis for specification development;

FIG. 24 is a flow diagram illustrating utilizing the site based highorder shape analysis for an unpatterned wafer geometry control process;

FIG. 25 is a flow diagram illustrating utilizing the site based highorder shape analysis for process uniformity control;

FIG. 26 is a block diagram illustrating a system for inspecting a waferin accordance with the present disclosure;

FIG. 27 is a flow diagram illustrating a polar grid partitioning methodin accordance with the present disclosure; and

FIG. 28 is a flow diagram illustrating another polar grid partitioningmethod in accordance with the present disclosure.

DETAILED DESCRIPTION

Reference will now be made in detail to the subject matter disclosed,which is illustrated in the accompanying drawings.

The present disclosure is directed to systems and methods for improvedresults of wafer higher order shape (HOS) characterization and waferclassification based on localized shapes. In accordance with the presentdisclosure, a wafer map is partitioned into a plurality of measurementsite areas to improve the completeness of wafer shape representation.This method may therefore be referred to as the site based high ordershape analysis method.

FIG. 1 is a flow diagram illustrating the major steps of the site basedhigh order shape analysis method 100 in accordance with the presentdisclosure. Data acquisition and HOS recipe may be created in step 102.Data acquisition and HOS recipe may specify how various wafer surfacemaps will be partitioned in later steps. Step 104 may acquire the wafersurface images directly utilizing wafer dimensional geometry tools suchas the WaferSight metrology system from KLA-Tencor. It is contemplated,however, that the wafer shape image, wafer front and back surface shapeimages or the like may also be constructed indirectly using othermetrology tools as well. Subsequently, step 106 may partition the wafermap into a plurality of measurement site areas and step 108 maycalculate HOS metrics for each of the plurality of measurement siteareas based on the wafer shape information (e.g., wafer shape, frontand/or back surface, image maps or the like) obtained in step 104. Step110 may then report the site based HOS metrics and may alsogroup/classify sites and wafers according to automatic or manually setthresholds.

More particularly, the recipe for site based high order shape may becreated based on Cartesian grid partition or polar grid partition. FIG.2 is an illustration depicting a Cartesian grid partition. It iscontemplated that different site sizes and shifts of the site array maybe selected according to requirements of desired spatial resolution andalignment with other measurement or process setup. As illustrated in thefigure, all sites in the Cartesian wafer surface partition have the samesite area, except the partial sites at wafer edge regions.

Alternatively, the wafer surface may also be partitioned into polar gridfor HOS analysis. Polar grid partition provides a better wafer edgeregion coverage. FIG. 3 is an illustration depicting an exemplary polargrid partition having 6 zones and 6 sectors. In this partition scheme,all polar sites have the same radial zone length L and the same sectorangular span θ. The zone boundaries of the zone may be determined byequation:

r _(i) =iL and r _(i-1)=(i−1)L, i=1,2,3, . . . K

For instance, the first zone is a circular-shaped zone with radius r₁=L,the second zone is a ring-shaped zone defined by an outer region havingan outer radius r₂=2 L and excluding an inner region having an innerradius r₁=L and so on. Furthermore, the area A of the polar site may bedetermined by equation:

$\begin{matrix}{{A_{i} = {{\frac{\theta}{2}\left( {r_{i}^{2} - r_{i - 1}^{2}} \right)} = {\frac{\theta}{2}\left( {i^{2} - \left( {i - 1} \right)^{2}} \right)L^{2}}}},} & {{i = 1},2,3,{\ldots \mspace{14mu} K}}\end{matrix}$

where K is the number of zones in the polar partition and KL=R is thewafer radius.

While this exemplary polar grid partition may be utilized for HOSanalysis, it is clear that the site areas defined by this partitionscheme may vary greatly. For example, in the scheme shown in FIG. 3, theratio of the site areas in the wafer edge and in the wafer center is 11.Such variations in site areas can result in the big spread in the HOSmeasurement values and affect the accurate wafer shape characterization.

The present disclosure therefore provides new polar grid partitionschemes that are able to partition the wafer surface into uniform areas.The first polar grid partition scheme in accordance with the presentdisclosure adopts the non-uniform radial span and defines the partitionand defines the site zone boundaries of the zone as:

r _(i) =√{square root over (i)}L, and r _(i-1)=√{square root over(i−1)}L=1,2,3, . . . K

where L is determined by the wafer radius R and the maximum zone numberK in the partition as:

$L = \frac{R}{\sqrt{K}}$

In accordance with the polar grid partition scheme described above, thesector angular span θ remains constant and the radial span for each zonemay vary to keep the site areas uniform. For example, for the samenumbers of the sectors and zones as depicted in FIG. 3, the polar sitescreated according to this new partition scheme are illustrated in FIG.4. It is noted that all polar sites now have the same areas and can beused to improve the accuracy of the wafer shape analysis.

Alternatively, the second polar grid partition scheme in accordance withthe present disclosure may adjust the angular span θ of the site in eachzone radius to obtain the uniform site area while keeping the radiallength constant. In this case, the angular span for the radial zone bandmay be determined utilizing equation:

$\begin{matrix}{{\theta_{i} = {\frac{1}{{2i} - 1}\frac{2\pi}{M}}},} & {{i = 1},2,3,{\ldots \mspace{14mu} K}}\end{matrix}$

where M is the number of the angular segments in the wafer center region(identified as region 502 in FIG. 5) and K is the number of the zones.

Using the second partition scheme for the case M=6 and K=6, the polarsite partition with uniform site areas can be obtained as shown in FIG.5, which has good wafer edge region coverage, good site spatialresolution and provides area uniformity.

It is contemplated that similar to the two schemes described above whereuniform site area is maintained by either keeping the radial lengthconstant and varying the angular span or vice-versa, uniform site areapolar partition may also be obtained by varying both the radial lengthand the angular span simultaneously. Furthermore, the polar gridpartitions having 6 zones and 6 sectors as described in the exampleabove are used merely for illustrative purposes. It is contemplated thatthe number of zones and the number of sectors may vary without departingfrom the spirit and scope of the present disclosure.

As illustrated in FIG. 1, once the wafer map is partitioned and thewafer shape information is obtained, step 108 may calculate HOS metricsfor each of the plurality of uniform measurement site areas based on theobtained wafer shape information. More specifically, given the siteimage shape data, I(x,y), in the Cartesian wafer partition as shown inFIG. 6A, the following best fitting surfaces of orders 0 to 2 may becalculated as defined by the following three equations, where N is thenumber of valid pixels in the given site image and the surfacecoefficients C(i,j) in these equations may be determined by the leastmean squared error (LMS) method. For instance, the mean level of siteimage may be calculated as:

$L_{C} = {\sum\limits_{({x,y})}\frac{I\left( {x,y} \right)}{N}}$

In addition, the first order best-fit surface of site image may becalculated as:

P _(c)(x,y)=C(0,0)+C(1,0)x+C(0,1)y

And the second order best-fit surface of site image may be calculatedas:

S _(c)(x,y)=C(0,0)+C(1,0)x+C(0,1)y+C(2,0)x ² +C(1,1)xy+C(0,2)y ²

It is contemplated that higher order best-fit surface of site image(order greater than 2) may be calculated to characterize the complexwafer surface geometry. In addition, the corresponding non-correctableshape components for different surface fitting orders may also becomputed to correlate to higher order process parameters.

For example, the deviations of the input site image, I(x,y), from thesite level L_(c) and two best fit surfaces, P_(c)(x,y) and S_(c)(x,y),may be calculated as:

D ₀(x,y)=I(x,y)−L _(c)

D ₁(x,y)=I(x,y)−P _(c)(x,y)

D ₂(x,y)=I(x,y)−S _(c)(x,y)

These deviation images (D₀, D₁ and D₂) are obtained by subtracting thepolynomial-fit surface from the original surface of each measurementsite. They represent the higher order shape components which cannot bedescribed by the corresponding zero order, first order and second ordersurface equations, and therefore cannot be corrected by thecorresponding zero order, first order and second order surfacecorrection processes. These various deviation metrics may also bereferred to as residues or shape residues, and variousdeviations/residues may be obtained by varying the order of the fittingpolynomial. Together with the surface coefficients, these deviationimages provide rich information about the wafer shape and can be used tocharacterize and sort the wafers effectively.

For instance, wafer shape information (may be referred to as surfaceshape metrics) that can be calculated for each measurement site based onthe surface coefficients may include: X Slope=C(1,0), which representsthe average site image slope in x direction with unit nm/mm; YSlope=C(0,1), which represents the average site image slope in ydirection with unit nm/mm;

${T_{1} = \sqrt[2]{{C^{2}\left( {1,0} \right)} + {C^{2}\left( {0,1} \right)}}},$

which represents the magnitude of the site image slope with unit nm/mm;and

${T_{2} = \sqrt[2]{{C^{2}\left( {2,0} \right)} + {C^{2}\left( {1,1} \right)} + {C^{2}\left( {0,2} \right)}}},$

which represents the magnitude of the second order surface componentswith unit nm/mm². It is noted that the magnitude of the first orderpolynomial fit coefficients is the magnitude of the shape-slope, and themagnitude of the second order polynomial fit coefficients is themagnitude of shape-curvature or simply shape-curvature. It iscontemplated that magnitude of other higher-order polynomial fitcoefficients may be derived in a similar manner.

Additional site slope metrics may also be derived from the surfacecoefficients and the site position angle φ (as depicted in FIG. 2)determined by the site center position on the wafer surface. Forinstance, the radial slope of the measurement site may be calculated asRadial Slope=C(1,0)×cos(φ)+C(0,1)×sin(φ) with unit nm/mm, and thetangential slope of the measurement site may be calculated as TangentialSlope=−C(1,0)×sin(φ)+C(0,1)×cos(φ) with unit nm/mm.

While the magnitudes of the shape slope and the shape curvature aredefined in equations above, it is contemplated that if more detailedsecond order components are required, they may be obtained from thethree surface components C(2,0), C(1,1) and C(0,2), which provide thesecond order shape curvature descriptions about the local shape. Forinstance, the curvature in x direction may be obtained as XCurvature=C(2,0), the curvature in y direction may be obtained as YCurvature=C(0,2), and the curvature in the (x=y) direction may beobtained as XY Curvature=C(1,1).

Furthermore, the following deviation metrics may also be constructedfrom the deviation images with unit nm:

P D₀ = max [D₀(x, y)]; V D₀ = min [D₀(x, y)];${{P\; V\; D_{0}} = {{P\; D_{0}} - {V\; D_{0}}}};{{M\; D_{0}} = {\sum\limits_{({x,y})}\frac{{D_{0}\left( {x,y} \right)}}{N}}}$P D₁ = max [D₁(x, y)]; V D₁ = min [D₁(x, y)];${{P\; V\; D_{1}} = {{P\; D_{1}} - {V\; D_{1}}}};{{M\; D_{1}} = {\sum\limits_{({x,y})}{{{D_{1}\left( {x,y} \right)}}/N}}}$P D₂ = max [D₂(x, y)]; V D₂ = min [D₂(x, y)];${{P\; V\; D_{2}} = {{P\; D_{2}} - {V\; D_{2}}}};{{M\; D_{2}} = {\sum\limits_{({x,y})}{{{D_{2}\left( {x,y} \right)}}/N}}}$

It is contemplated that these deviation metrics provide the informationabout the wafer shape after certain correction procedures. For example,the metric PD₀ tells the maximum positive error after the site levelingoperation, the metric MD₁ denotes the average deviation error after thepiston/tilt correction, which may be carried out by the stepper/scannerin auto-focus process, and MD₂ gives the information about the surfacecomponents higher than the second order.

FIG. 7 illustrates one original site shape image I(x,y) and itscorresponding two deviation images D₁(x,y) and D₂(x,y) where thedistributions of the shape components in the original shape image can beclearly seen from two deviation images. It is noted that if the originalshape image has only shape components of the first order, D₁(x,y) willbe an all zero plane. Similarly, if the original image has only theshape components up to the second order, then the second order deviationimage D₂ (x,y) will be an all zero plane.

It is contemplated that the HOS metrics calculated for each of themeasurement site areas in step 108 may be utilized to group/classify thesite areas for reporting purposes in step 110. In addition, automatic ormanually set thresholds may be utilized to visualize the site based highorder shape analysis results. For example, FIG. 8 shows a wafer shapeimage and its X profile. This wafer has high shape slope at the waferedge region and has higher order components than linear terms at thewafer radius of 40 mm region. The X slope metric map, the Y slope metricmap, the radial slope map and the tangential slope map may be calculatedaccording to the equations described above and the calculated sitemetric values may be presented for each measurement site in acorresponding map as shown in FIG. 9. Alternatively/additionally, thecalculated shape metrics and the classification results may also bereported in measurement result files (e.g., a text-based ormachine-readable result file).

In the maps shown in FIG. 9, each site may be shaded (or colored basedon the specific implementation) according to the calculated site metricvalues and shading/coloring rules. For instance, a site may beshaded/colored in a first manner if its metric value is below a lowerthreshold, in a second manner if its metric value is between the lowerthreshold and an upper threshold, or in a third manner if its metricvalue is above the upper threshold. It is contemplated that the lowerand upper thresholds may be set manually by the user to classify thesites and wafers. Alternatively, the thresholds may be determinedautomatically. For instance, a median value of absolute metric values ofall sites may be calculated first and used as the lower threshold. Theupper threshold may be defined subsequently as twice the lowerthreshold. It is understood, however, that utilizing two thresholds ismerely exemplary. The number of thresholds utilized forgrouping/classifying the measurement sites may vary without departingfrom the spirit and scope of the present disclosure. Furthermore, thethreshold values may also be determined differently as described above.

It is clear that the four slope maps shown in FIG. 9 describe the wafershape slope in different orientations and provide comprehensive wafershape information. The radial slope map and the tangential slope mapindicate that wafer shape has bigger slope in the wafer edge region thanthe interior region and that the wafer shape has very smooth shapeprofile in the tangential direction. Furthermore, to obtain shapeinformation of higher order components, the metrics from the first orderdeviation map, such as PVD₁, or the metrics from the second orderfitting coefficients, such as T₂, may be utilized. FIG. 10 shows thedeviation PVD₁ map and the second order coefficient derived T₂ map. Bothof them characterize the wafer shape components higher than the firstorder and they both show that there are bigger higher order componentsin wafer radius of 40 mm neighborhood region. T₂ metric map alsoindicates that there are higher order shape components in wafer edgesites.

FIG. 11 shows another exemplary wafer shape image and its profile in−45° orientation. The corresponding four slope metric maps are shown inFIG. 12. As indicated in FIG. 12, the high shape slope in the south-eastwafer edge region is well detected by these slope metric maps. Inaddition, the tangential slope map shows that this wafer has highertangential slope components in the 3 o'clock and 6 o'clock wafer edgeregions than other wafer edge regions. Furthermore, FIG. 13 shows thePVD, map, T₂ map and PVD₂ map, where the PVD₂ map clearly shows that thedeep shape valley in the wafer center region has the shape componentshigher than the second order. It is contemplated that providing theability for the user to visualize the information as depicted in FIGS.12 and 13 is appreciated in various situations. However, it is alsocontemplated that such information may be reported in a classificationresult file (e.g., a text-based or machine-readable result file) withoutdeparting from the spirit and scope of the present disclosure.

While the surface fitting coefficients and deviation images discussedabove are defined on the Cartesian (x,y) image site partitions, theprinciples and methods can be extended to polar (r,β) partitions aswell. For instance, if polar position is used for the HOScharacterization, the acquired wafer image maps (usually in Cartesianspace) may be converted/transformed into polar space first. The entirewafer polar image may then be partitioned into rectangular polar spacedata blocks I(r,β) as shown in FIG. 6B. The boundaries of these polarspace data blocks are defined by the particular partition schemeselected (e.g., from the schemes described in FIG. 3, 4 or 5). That is,the polar site image in FIG. 6B corresponds to a site area in FIG. 3, 4or 5 based on the partition scheme selected, and the polar sites shownin Cartesian space in FIGS. 3, 4 and 5 becomes rectangular data blocksin the polar space depicted in FIG. 6B.

The surface fitting and deviation image calculation may now becalculated as follows:

$\mspace{20mu} {L_{p} = {\sum\limits_{({r,\beta})}\frac{I\left( {r,\beta} \right)}{N}}}$  P_(p)(r, β) = C(0, 0) + C(1, 0)r + C(0, 1)βS_(p)(r, β) = C(0, 0) + C(1, 0)r + C(0, 1)β + C(2, 0)r² + C(1, 1)r β + C(0, 2)β²

Similarly, the deviations of the input site image, I(r,β), from the sitelevel L_(p) and two best fit surfaces, P_(p)(x,y) and S_(p)(r,β) may becalculated as:

D ₀(r,β)=I(r,β)−L _(p)

D ₁(r,β)=I(r,β)−P _(P)(r,β)

D ₂(r,β)=I(r,β)−S _(P)(r,β)

Additional site slope metrics may also be derived from the surfacecoefficients. For instance, the radial slope of the measurement site maybe calculated as Radial Slope=C(1,0), which represents the average siteimage shape slope in r direction, and the tangential slope of themeasurement site may be calculated as Tangential Slope=C(0,1), whichrepresents the average site image shape slope in β direction.Furthermore, the X shape slope and Y shape slope values in the polarspace partition may be calculated from the radial shape slope,tangential shape slope and the site center angle Ω (as depicted in FIG.5). For instance, the X shape slope may be calculated as X Slope=RadialSlope*cos(Ω)−Tangential Slope*sin(Ω) and the Y shape slope may becalculated as Y Slope=Radial Slope*sin(Ω)+Tangential Slope*cos(Ω). Othermetrics such as T₁ and T₂ may also be calculated in a similar manner asdescribed above. T₁ and T₂ provide the magnitude of the shape slope andthe magnitude of the second order shape components or shape curvaturecomponents.

Similar to the Cartesian space described above, the HOS metricscalculated for each of the measurement site areas in the polar space mayalso be utilized to group/classify the site areas for reportingpurposes. For example, FIG. 14 shows the X slope metric map, the Y slopemetric map, the radial slope map and the tangential slope map for anexemplary wafer partitioned utilizing polar partitioning. It isunderstood that the polar space partitioning scheme depicted in FIG. 14is merely exemplary. The uniform site area partitioning scheme depictedin FIGS. 4 and 5 may also be utilized for reporting purposes withoutdeparting from the spirit and scope of the present disclosure.Furthermore, it is contemplated that polar partition may provide betterwafer shape characterization and better correlation with the processinformation in certain applications.

Alternative to calculating the HOS metrics based on fitting first orderpolynomials across sites/polar-sectors as described above, anothertechnique for shape slope computation is to compute the X slope, the Yslope, the radial slope and the tangential slope (may be jointlyreferred to as x/y/radial/tangential components) at every pixel locationvia numerical methods such as forward difference, backward difference,and central difference methods. For instance, subsequently to capturelocal shape-slope effects efficiently and report to the user in the formof images and text-based data output, the pixel-based shape-slope maps(x/y/radial/tangential components) may be segmented intosites/polar-sectors and a mean value of shape-slopes(x/y/radial/tangential components) may be reported for eachsite/polar-sector. Similarly max/min/range and other values may bereported per site/polar-sector. For illustration purpose a contour imageof the mean radial shape slope for a wafer segmented by sites is shownin FIG. 15A. This is also compared to slope computation by way ofpolynomial fitting described earlier in and as shown in FIG. 15B.

It is observed that the two methods of slope computation (i.e., based onfitting first order polynomials, or alternatively, based on numericalmethods) produce very similar results. It is therefore contemplated thatstep 108 may utilize either method to calculate metrics for themeasurement sites. It is further contemplated that other alternativecomputation methods may also be utilized to compute the various metricsdescribed above. The specific equations and/or method utilized may varywithout departing from the spirit and scope of the present disclosure.

It is also observed that the metrics described above provide metricvalues for each field/site. These metric values by definition aresuitable for performing inter-field (field-to-field variations) dataanalysis, but may not be optimal for performing intra-field (withinfield variations) data analysis. It is contemplated, however, that themethod in accordance with the present disclosure may be adapted toprovide metrics for multiple data points (may be referred to as targets)per measurement site. Providing metrics for multiple data points foreach site will therefore support intra-field data analysis, which may beappreciated in various wafer measurement applications.

FIG. 16 describes a new methodology for performing intra-field dataanalysis. Step 1602 may receive the wafer shape data as input and step1604 may compute the shape slope data by pixel level computation or bypolynomial fitting computation method as described above. Using theexample of lithography overlay variation, overlay errors are typicallymeasured at a number of pre-determined target locations within alithography field and across multiple fields within a device wafer. Thisis shown in FIG. 17. These target locations across the wafer 1702 may befed back to the wafer geometry tool in order to be used as samplinglocations for shape-slope metric or shape-slope residual metricmeasurement. The development and usage of shape-slope residual metric isdescribed in: Overlay and Semiconductor Process Control Using a WaferGeometry Metric, P. Vukkadala et al., U.S. patent application Ser. No.13/476,328, which is herein incorporated by reference in its entirety.Furthermore, the feed-back loop 1802 for feeding process measurelocations to the wafer geometry tool 1804 is illustrated in FIG. 18,which depicts a wafer lithography process relative to a timeline.

Step 1606 may then obtain the shape slope data and other higher-ordershape (HOS) data for certain user specified target locations and step1608 may utilize the shape slope data at these target locations to studyintra-field lithography process variation such as overlay variation. TheHOS values measured at these target locations may then be utilized toperform intra-field data analysis in step 1610. For instance, step 1610may compare the HOS values measured at these target locations to processdata such as overlay errors (using visual color maps or statisticalcorrelation analysis) in order to identify the correlation between HOSvalues and process variation. Such analysis can be used to assess theimpact of HOS on intra-field process variation.

It is contemplated that alternative approaches may also be utilized forassessing the impact of HOS on inter-field process variations. Forinstance, the process data such overlay data (measured at severaltargets per field and at multiple fields across the wafer) may bepartitioned and re-formatted into sites (fields) and sectors exactlysimilar to the partitioning scheme used with the corresponding wafergeometry data. Thus metrics such as mean overlay, peak-to-valley overlayand the like may be computed per field/sector for multiplefields/sectors across the wafer. This may then be compared to site-basedor sector-based wafer geometry metrics to assess the impact of wafergeometry variation (HOS) on inter-field process variation.

It is also contemplated that the site based high order shape analysismethod and system in accordance with the present disclosure may beappreciated in various other wafer analysis applications. For example,the various HOS metrics described above may be utilized to control aChemical Mechanical Planarization or Polishing (CMP) process.

More specifically, modeling simulation results reported in P. Vukkadalaet al., “Impact of Wafer Geometry on CMP for Advanced Nodes,” Journal ofElectrochemical Society (JES), Vol. 158, No. 10, pp. H1002-H1009, 2011,shows that the uniformity of CMP processes such as Shallow TrenchIsolation (STI) are highly dependent on the higher order components ofthe shape of a wafer. This is illustrated in FIG. 19, which showsincreasing amount of nitride thinning experienced as the magnitude ofwafer lip (a higher order shape component) increases. A typical methodfor measuring nitride thinning during a CMP process is by measuring theSTI step height. The process data such as STI step height variation aretypically measured at finite number of points on the surface of thewafer. To assess the correlation between the process data and the wafergeometry metrics, the process data needs to be formatted appropriately.The methodology for formatting the process data is dependent on theformat of the wafer geometry metrics. For example, for comparing thesector metrics the process data will be grouped into various sectors andrelevant metrics will be computed on the process data.

Consequently experiments were conducted to assess the impact of higherorder shape on CMP removal uniformity. It was determined that the RadialShape-Slope metric (both sites/polar-sector based) correlated well withthe STI step height variation process data. Hence the Radial Shape Slopemetric of an unpatterned/filmed wafer may be used to control theuniformity of CMP processes such as STI. This may be achieved by havingan inline monitor for Radial Shape Slope to assess the amount of CMPnon-uniformity an incoming wafer may exhibit down the line after a CMPprocess.

This is illustrated in FIG. 20 where the wafer geometry (including HOS)is measured at the bare/unpatterned wafer level as well as severalprocess steps later but right before the wafer is subjected to CMP. Thisway two things may be determined: (i) by computing the difference inhigher order shape between the two wafer geometry measurements gives anindication of the amount of higher order shape induced by waferprocessing; and (ii) by correlating higher order shape (e.g., RadialShape Slope) to CMP removal variation across the wafer a model may bedeveloped for predicting the amount of CMP variation that may be causedby a given wafer higher order shape. Proper thresholds for the waferhigh order shapes may be developed to limit the amount of CMP variationto an acceptable level. Thus an inline wafer geometry (higher ordershape) monitor may be used to: (i) accept/reject a wafer for aparticular process, (ii) identify process steps that induce largerhigher order shape for further root-cause-analysis, and (iii) sortincoming wafers into technology node specific bins.

Another example of the application of wafer site based higher ordershape metrics is to monitor the impact of wafer shape on lithographyprocess. During patterning a wafer using lithography process, the waferis first held on a vacuum or electrostatic chuck (based on thelithography technology) by using vacuum or electrostatic forcerespectively. When the wafer is held on a chuck using a force, theinitial gap between the wafer and chuck primarily due to the shape ofthe wafer is reduced. Ideally the wafer back surface is expected tocompletely come in contact with the chuck surface with zero contact gap.However, in reality the contact gap is a function of the wafer geometry.Contact gap may result in defocus errors and need to be monitored andcontrolled. Previously, there was no metric to monitor the contact gapduring chucking.

FIG. 21 illustrates the change of the contact gap(s) as a result ofchucking. Consequently, a three-dimensional finite element model withtens of thousands of nodes may be developed to simulate the wafer andchuck interaction. The key input parameters of the model may include thewafer geometry, the chuck geometry, and the applied vacuum/electrostaticpressure. This model may be developed assuming uniform chucking pressureover the entire surface of the wafer. One of the key outputs of themodel may include contact gap estimation. With the pressure and chuckgeometry being constant, the impact on different wafer geometry oncontact gap may be estimated/observed. Experimental results haveindicated that contact gap is a function of the curvature of wafer shapeand a good correlation was observed between contact gap and the sitebased shape curvature metric as shown in FIG. 22. Therefore, the sitebased shape curvature metrics provided utilizing the method and systemof the present disclosure may be utilized to monitor/access the impactof wafer shape on lithography process.

In addition to utilizing the HOS metrics to control a CMP andlithography process, it is contemplated that the HOS metrics may beutilized for controlling other processes without departing from thespirit and scope of the present disclosure. For instance, FIG. 23 is aflow diagram depicting utilizing HOS metrics for specificationdevelopment, FIG. 24 is a flow diagram depicting an unpatterned wafergeometry control process, and FIG. 25 is a flow diagram depictingprocess uniformity control, all based on the HOS metrics described inaccordance with the present disclosure. It is also contemplated that theHOS metrics may be utilized for process control for mitigating overlayerrors as well as other wafer analysis/control applications.

FIG. 26 is a block diagram depicting a wafer inspection system 2600 inaccordance with the present disclosure. The wafer inspection system 2600includes an optical system 2602 configured for obtaining a wafer surfaceimage. As previously described, the optical system 2602 may acquire thewafer surface images directly utilizing wafer dimensional geometry toolssuch as the WaferSight metrology system from KLA-Tencor. Alternatively,the wafer shape image, wafer front and back surface shape images or thelike may also be constructed indirectly using other metrology tools aswell.

The wafer inspection system 2600 also includes a site based high orderwafer shape analysis module 2604 in communication with the opticalsystem 2602. The site based high order wafer shape analysis module 2604is configured for carrying out the site based high order shape analysismethod 100 as described above. The calculated high order shape metricsmay subsequently be utilized as control input for various downstreamapplications 2606, including, but not limited to, CMP processes, waferspecification development processes, unpatterned wafer geometry controlprocesses, wafer uniformity control processes or the like.

FIG. 27 is a flow diagram illustrating the polar grid partitioningmethod 2700 in accordance with the present disclosure. Step 2702 mayspecify a number of sectors and a number of zones required for the polargrid partitioning. Step 2704 may calculate a sector angular span basedon the number of sectors specified. Step 2706 may calculate a radialspan for each of the number of zones. In accordance with thispartitioning scheme, the radial span for a first zone having a firstradial distance to the center of the wafer is different from the radialspan for a second zone having a second radial distance to the center ofthe wafer. Step 2708 may then partition the wafer surface into aplurality of sites based on the sector angular span and the radial spanfor each zone. The sites partitioned in this manner will have uniformsite areas.

FIG. 28 is a flow diagram illustrating an alternative polar gridpartitioning method 2800 in accordance with the present disclosure. Step2802 may specify a number of zones K required for the polar gridpartitioning and a number of angular segments M in a center region ofthe wafer. Step 2804 may calculate a radial zone length L based on thenumber of zones specified. Step 2806 may independently calculate anangular span θ_(i) for the i^(th) radial zone, wherein i=1, 2, 3, . . .K. Step 2808 may subsequently partition the wafer surface into aplurality of uniform sites based on the radial zone length L and theangular span θ for each radial zone. The sites partitioned in thismanner will have uniform site areas.

It is contemplated that while the examples above referred to wafermetrology measurements, the systems and methods in accordance with thepresent disclosure are applicable to other types of polished plates aswell without departing from the spirit and scope of the presentdisclosure. The term wafer used in the present disclosure may include athin slice of semiconductor material used in the fabrication ofintegrated circuits and other devices, as well as other thin polishedplates such as magnetic disc substrates, gauge blocks and the like.

The methods disclosed may be implemented as sets of instructions,through a single production device, and/or through multiple productiondevices. Further, it is understood that the specific order or hierarchyof steps in the methods disclosed are examples of exemplary approaches.Based upon design preferences, it is understood that the specific orderor hierarchy of steps in the method can be rearranged while remainingwithin the scope and spirit of the disclosure. The accompanying methodclaims present elements of the various steps in a sample order, and arenot necessarily meant to be limited to the specific order or hierarchypresented.

It is believed that the system and method of the present disclosure andmany of its attendant advantages will be understood by the foregoingdescription, and it will be apparent that various changes may be made inthe form, construction and arrangement of the components withoutdeparting from the disclosed subject matter or without sacrificing allof its material advantages. The form described is merely explanatory.

What is claimed is:
 1. A method for inspecting a wafer, comprising:defining a wafer partitioning scheme; obtaining a wafer surface image;partitioning the wafer surface image into a plurality of measurementsites according to the wafer partitioning scheme; calculating aplurality of measurement metrics for each of the plurality ofmeasurement sites based on the acquired wafer surface image; andreporting the plurality of measurement metrics calculated for each ofthe plurality of measurement sites in a graphical representation.
 2. Themethod of claim 1, further comprising: reporting the plurality ofmeasurement metrics calculated for each of the plurality of measurementsites in a measurement result file.
 3. The method of claim 1, furthercomprising: classifying the plurality of measurement sites according toat least one of: a manually set threshold or an automatically determinedthreshold; and reporting the classification results in at least one of:a graphical representation or a classification result file.
 4. Themethod of claim 1, wherein the wafer partitioning scheme is a Cartesiangrid partition scheme, the Cartesian grid partition scheme partitionsthe measurement sites into a plurality of uniform rectangular siteareas.
 5. The method of claim 1, wherein the wafer partitioning schemeis a polar grid partition scheme.
 6. The method of claim 5, wherein themeasurement sites partitioned according to the polar grid partitionscheme have non-uniform site areas.
 7. The method of claim 5, whereinthe measurement sites partitioned according to the polar grid partitionscheme have uniform site areas.
 8. The method of claim 7, wherein theuniform site area is provided by adjusting at least one of: a radialspan of a zone, or an angular span of a sector.
 9. The method of claim1, wherein the measurement metrics calculated for each measurement siteincludes at least one of: a plurality of surface shape metrics obtainedby fitting polynomial equations to the surface of the measurement site;and a plurality of deviation metrics obtained by subtracting thepolynomial-fit surface from the original surface of the measurementsite.
 10. The method of claim 9, wherein the surface shape metrics foreach measurement site includes at least one of: an average slope of themeasurement site in x direction; an average slope of the measurementsite in y direction; a magnitude of the measurement site slope; amagnitude of a second order surface components for the measurement site;a radial slope of the measurement site; and a tangential slope of themeasurement site.
 11. The method of claim 9, wherein the measurementmetrics are calculated based on surface coefficients obtained utilizingat least one of: a polynomial fitting process or a pixel-basedshape-slope computation process.
 12. The method of claim 1, wherein themeasurement metrics are utilized as control inputs for controlling atleast one of: a Chemical Mechanical Planarization or Polishing (CMP)process, a wafer specification development process, an unpatterned wafergeometry control process, or a wafer uniformity control process.
 13. Themethod of claim 1, wherein the surface shape metrics for eachmeasurement site further includes shape curvature information, andwherein the shape curvature information is utilized to assess an impactof a contact gap occurred during clamping of the wafer on a lithographychuck.
 14. The method of claim 1, further comprising: receiving aplurality of specified target locations; obtaining the plurality ofmeasurement metrics for each of the plurality of target locations; andperforming intra-field data analysis based on the plurality ofmeasurement metrics obtained for each of the plurality of targetlocations.
 15. A system for inspecting a wafer, comprising: an opticalsystem configured for obtaining a wafer surface image; a site based highorder wafer shape analysis module in communication with the opticalsystem, the site based high order wafer shape analysis module configuredfor: defining a wafer partitioning scheme; partitioning the wafersurface image into a plurality of measurement sites according to thewafer partitioning scheme; calculating a plurality of measurementmetrics for each of the plurality of measurement sites based on theacquired wafer surface image; and reporting the plurality of measurementmetrics calculated for each of the plurality of measurement sites in agraphical representation.
 16. The system of claim 15, wherein the sitebased high order wafer shape analysis module is further configured for:reporting the plurality of measurement metrics calculated for each ofthe plurality of measurement sites in a measurement result file.
 17. Thesystem of claim 15, wherein the site based high order wafer shapeanalysis module is further configured for: classifying the plurality ofmeasurement sites according to at least one of: a manually set thresholdor an automatically determined threshold; and reporting theclassification results in at least one of: a graphical representation ora classification result file.
 18. The system of claim 15, wherein thewafer partitioning scheme is a Cartesian grid partition scheme, theCartesian grid partition scheme partitions the measurement sites into aplurality of uniform rectangular site areas.
 19. The system of claim 15,wherein the wafer partitioning scheme is a polar grid partition scheme.20. The system of claim 19, wherein the measurement sites partitionedaccording to the polar grid partition scheme have non-uniform siteareas.
 21. The system of claim 19, wherein the measurement sitespartitioned according to the polar grid partition scheme have uniformsite areas.
 22. The system of claim 21, wherein the uniform site area isprovided by adjusting at least one of: a radial span of a zone, or anangular span of a sector.
 23. The system of claim 15, wherein themeasurement metrics calculated for each measurement site includes atleast one of: a plurality of surface shape metrics obtained by fittingpolynomial equations to the surface of the measurement site; and aplurality of deviation metrics obtained by subtracting thepolynomial-fit surface from the original surface of the measurementsite.
 24. The system of claim 23, wherein the surface shape metrics foreach measurement site includes at least one of: an average slope of themeasurement site in x direction; an average slope of the measurementsite in y direction; a magnitude of the measurement site slope; amagnitude of a second order surface components for the measurement site;a radial slope of the measurement site; and a tangential slope of themeasurement site.
 25. The system of claim 23, wherein the measurementmetrics are calculated based on surface coefficients obtained utilizingat least one of: a polynomial fitting process or a pixel-basedshape-slope computation process.
 26. The system of claim 15, wherein themeasurement metrics are utilized as control inputs for controlling atleast one of: a Chemical Mechanical Planarization or Polishing (CMP)process, a wafer specification development process, an unpatterned wafergeometry control process, or a wafer uniformity control process.
 27. Thesystem of claim 15, wherein the surface shape metrics for eachmeasurement site further includes shape curvature information, andwherein the shape curvature information is utilized to assess an impactof a contact gap occurred during clamping of the wafer on a lithographychuck.
 28. The system of claim 15, wherein the site based high orderwafer shape analysis module is further configured for: receiving aplurality of specified target locations; obtaining the plurality ofmeasurement metrics for each of the plurality of target locations; andperforming intra-field data analysis based on the plurality ofmeasurement metrics obtained for each of the plurality of targetlocations.
 29. A polar grid partitioning method for partitioning a wafersurface, the method comprising: specifying a number of sectors and anumber of zones required for the polar grid partitioning; calculating asector angular span based on the number of sectors specified; utilizinga computer or processor to calculate a radial span for each of thenumber of zones, wherein the radial span for a first zone having a firstradial distance to the center of the wafer is different from the radialspan for a second zone having a second radial distance to the center ofthe wafer; and partitioning the wafer surface into a plurality of sitesbased on the sector angular span and the radial span for each zone,wherein the plurality of sites have uniform site areas.
 30. The polargrid partitioning method of claim 29, wherein the radial span for thefirst zone is greater than the radial span for the second zone when thefirst radial distance is smaller than the second radial distance. 31.The polar grid partitioning method of claim 29, wherein the radial spanfor the radial zone is defined as the span between boundaries:r _(i) =√{square root over (i)}L and r _(i-1)=√{square root over(i−1)}L, i=1,2,3, . . . K wherein ${L = \frac{R}{\sqrt{K}}},$ and R isthe radius of the wafer and K is the number of zones.
 32. A polar gridpartitioning method for partitioning a wafer surface, the methodcomprising: specifying a number of zones K required for the polar gridpartitioning and a number of angular segments M in a center region ofthe wafer; calculating a radial zone length L based on the number ofzones specified; utilizing a computer or processor to independentlycalculate an angular span θ_(i) for the i^(th) radial zone, wherein i=1,2, 3, . . . K; and partitioning the wafer surface into a plurality ofsites based on the radial zone length L and the angular span θ for eachradial zone, wherein the plurality of sites have uniform site areas. 33.The polar grid partitioning method of claim 32, wherein the angular spanθ_(i) is less than the angular span θ_(i-1) for i>1.
 34. The polar gridpartitioning method of claim 32, wherein the angular span θ_(i) for thei^(th) radial zone is calculated according to equation: $\begin{matrix}{{\theta_{i} = {\frac{1}{{2i} - 1}\frac{2\pi}{M}}},} & {{i = 1},2,3,{\ldots \mspace{14mu} {K.}}}\end{matrix}$